Introducing a Polynomial Expression of Molecular Integrals for Algebraic the Molecular Orbital (MO) Equation

Abstract

Abstract An algebraic molecular orbital (MO) equation is proposed, based on molecular integrals expressed as polynomials with respect to internal molecular coordinates and orbital exponents of basis functions. The algebraic MO equation deals with molecular coordinates as variables and orbital exponents as nonlinear variational parameters, which are not considered in the Hartree–Fock–Roothaan (HFR) equation. In this chapter, a method to express molecular integrals over Slater-type orbitals in polynomial form, polynomial function is demonstrated for atoms. Applying Taylor-series expansion to analytical molecular integrals, polynomial molecular integrals are obtained with accuracy being controlled by the order of Taylor-series and their coefficients expressed as rational numbers. The algebraic MO simultaneous equations is stem from first-order necessary conditions for local minima of total electronic energy, orthonormality of MOs and the constraints named the Schrodinger condition, such as Kato's cusp condition and the virial theorem. Solution of the algebraic MO equation is expected to include electron correlation because the solution of the Schrodinger equation should obey the Schrodinger condition. Polynomial molecular integrals with controlled accuracy define the algebraic MO equation with enough accuracy to solve multivariable problems in quantum chemistry.